Exam

BITSAT

Year

2022

Subject

Mathematics

Chapter

Area Under Curves

BITSAT 2022 · Mathematics · Area Under Curves

The area of the region bounded by the parabola (y-2)^2=(x-1) , the...

The area of the region bounded by the parabola $(y-2)^2=(x-1)$, the tangent to the parabola at the point $(2,3)$ and the $X$-axis is

Step-by-step solution

The equation of the parabola is $y^2-4 y-x+5=0$ The equation of tangent at $(2,3)$ is $$\begin{aligned}3 y-2(y+3)-\frac{x+2}{2}+5 & =0 \\ 2 y-x-4 & =0 \end{aligned}$$$\\$\therefore$Required area$A$is given by$$\begin{aligned}& A=\int_0^3\left(x_2-x_1\right) d y \\ & \Rightarrow \quad A=\int_0^3\left[\left\{(y-2)^2+1\right\}-\{2 y-4\}\right] d y \\ & \Rightarrow \quad A=\int_0^3\left(y^2-6 y+9\right) d y=\int_0^3(3-y)^2 d y \\ & =-\left[\frac{(3-y)^3}{3}\right]_0^3=9 \\ & \end{aligned}$$